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A Remark on the Blowing-Up of Solutions to the Cauchy Problem for Nonlinear Schrödinger Equations

O. Kavian
Transactions of the American Mathematical Society
Vol. 299, No. 1 (Jan., 1987), pp. 193-203
DOI: 10.2307/2000489
Stable URL: http://www.jstor.org/stable/2000489
Page Count: 11
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
A Remark on the Blowing-Up of Solutions to the Cauchy Problem for Nonlinear Schrödinger Equations
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Abstract

We consider solutions to iut = Δ u + |u|p-1, u(0) = u0, where x belongs to a smooth domain $\Omega \subset R^N$, and we prove that under suitable conditions on p, N and u0 ∈ H2(Ω) ∩ H1 0(Ω), |∇ u(t)|L2 blows up in finite time. The range of p's for which blowing-up occurs depends on whether Ω is starshaped or not. Examples of blowing-up under Neuman or periodic boundary conditions are given.

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